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Key on, engine off fuel pressure should be between 45 and 56 psi. Key on, engine running should be 50-56 psi. Have you measured fuel pressure when the car starts and runs fine? It's not uncommon for a failing fuel pump to exhibit failing symptoms as the pump heats up. Also, since the fuel pump is a DC motor located in the fuel tank, the fuel acts as a heat sink for the fuel pump. Often failing pumps will work better on a full tank of fuel than on a low tank. Of course they will certainly fail when the tank is over half full since they get pretty heavy at that point.
-Rod